Nonlinear Equations - UFRJ

148 [CH. 10: HOMOTOPY
above, the sequence of continuous functions h k (s) = N k (F s , X i ) is
uniformly convergent to Z s = lim k→∞ N k (F s , X i ). Hence, Z s is
continuous and is a representative of z s . Since [lim N k (F s , X i )] =
[lim N k (F s , X i+1 )], item 2 of the Theorem follows.
Now to item 3: except for the final step, every step in the algorithm
falls within two possibilities: either s = S 1 , or s = S 2 . Then
Lemma 10.12 and 10.13 say that
L(f t , z t ; t i , t i+1 ) ≥
1
CD 3/2√ n
Remark 10.14. The constants were computed using the free computer
algebra package Maxima [60] with 40 digits of precision, and checked
with 100 digits. The first thing to do is to guess a viable point
(α, ɛ 1 , ɛ 2 ) satisfying (10.3), (10.15), (10.17) and (10.18), for instance
(0.05, 0.02, 0.04).
Then, those values are optimized for min(ɛ 1 , ɛ 2 ) by adding a small
Gaussian perturbation, and discarding moves that do not improve the
objective function or leave the viable set. Slowly, the variance of the
Gaussian is reduced and the point converges to a local optimum. This
optimization method is called simulated annealing.
10.3 Average complexity of randomized
algorithms
In the sections above, we constructed and analyzed a linear homotopy
algorithm. Now it is time to explain how to obtain a proper starting
pair (F 0 , x 0 ).
Here is a simplified version of the Beltrán-Pardo construction of
a randomized starting system. It is assumed that our randomized
computer can sample points of N(0, 1). The procedure is as follows.
Let M be a random (Gaussian) complex matrix of size n × n + 1.
Then find a nonzero Z 0 ∈ ker M. Next, draw F 0 at random in the
subspace R M of H d defined by L Z0 (F 0 ) = M, F 0 (Z 0 ) = 0. This can
be done by picking F 0 at random, and then projecting.